String Theory and M-Theory | Lecture 7

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String Theory and M-Theory | Lecture 7
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(November 1, 2010) Leonard Susskind discusses the specifics of strings including Feynman diagrams and mapping particles. String theory (with its close relative, M-theory) is the basis for the most ambitious theories of the physical world. It has profoundly influenced our understanding of gravity, cosmology, and particle physics. In this course we will develop the basic theoretical and mathematical ideas, including the string-theoretic origin of gravity, the theory of extra dimensions of space, the connection between strings and black holes, the "landscape" of string theory, and the holographic principle.
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Steiger University OK let's
say let's go tool armed thee persist based in view of string theory but what kind of you could it be but space time but I mean that's let's think of the black border space time and view of what for string theory works like 1st something simple 1st let's just talk about ordinary particle an ordinary particle in classical mechanics has a trajectory in unreliable both in nonrelativistic physics and relativistic physics the trajectory can be described by the the three-dimensional position X as a function of but if you doing relativity something which is more convenient and that's the take all 4 components space components and time components are position call X you and think of them as a function other variables which just being notes being marks a proper time along the trajectory now but how the proper time along with trajectory and they clearly if you know the four-dimensional position at every proper time you are every every value of this parameter here then you could say exactly what the trajectory where some of us like will only be motorists like a difference of emphasis in nonrelativistic physics where you think of X is a function of time and in relativistic physics will you might be inclined to think of X knew 4 components a position as a function of a parameter along the trajectories scalar Parma along the trajectory in each case the motion our particle is defined by an action by a principle of stature in the case of a nonrelativistic system for a free particle forget potential nothing action which is usually called AS Roma whites called us saying label as we used entropy and to make matters worse in many cases entropy inaction turned out to be the same numerically S is action and for relativistic particle
state in all of in the world all the time along the trajectory of the kinetic energy one-half X stop squared and I've chosen the massive particle arbitrarily to be 1 because I get tired of writing that's the idea that the action don't you do with it you minimize minimize defined trajectory of particle you minimize subject for some 18 positions holding them in points you minimize no that's not what you doing quite quad or before we do that the to discuss the relativistic case the relativistic case something very similar the thing is they actually quake taken get the girl all right now the proper time along the trajectory of the X you'll by Cowell DD X you come notices when upper index and lower index you could figure out that were that no it's a lives very very much like the nonrelativistic kinetic energy formula but means something different for components of a position and the time is be the proper time and what's more but time components and space component of this expression have different sign but path and that they look very similar I love and again what you do with it you minimize you minimize the subject of the constraint that D & B ends of the periods of particles of this space time poring and that's pretty space Corey a sort apparel structures and basic principles principally stature quantum mechanics you do something totally different with action what you do it reacts different questions you don't ask What is the trajectory given to win points ADDS was a trajectory given to end points the answer is of course a straight line EDT case straight line straight line to space and I love with constant velocity US a different question you ask Wednesday is the input to the particle which started at this point at times 0 the beauty of Prime start time formidable is detected at that point at a later in the way the part that's a question less you don't ask What's the trajectory in between U.S. amplitude for from here and the insert it is the amplitude 0 1 1 2 . 2 will be the case you write the actions write the exponential reactor he be body not just any exponential but exponential what I eat so they are lying policy action was right are explicitly IDT over one-half treaty d X by D. C squared this integral here goes from T-1 52 but they had not finished you not finished this expression depends not only on the position of particle here now he but but it depends on the position of a particle everywhere between so what do you do with a defined the amplitude Europe is the greater use some what all possible trajectories somehow on all trajectories all are an integral strictly speaking all trajectories misses a monstrously complicated things it's not a simple integral it you could put on nurse or what his name young Mathematica left it's incident that maintenance grow continuously infinite image the rope thought that mathematical monstrosity of about this this it to go over here just a the along the trajectory that's easy as just action but this integral over here is the sum of all trajectories particle feels out all trajectories it doesn't sample the 1 of action you if the great although all of them this expression of finding amplitude go from 1 that's upon mechanical work a version of the principle of least action finding patentable you the same thing in either classical McCann agony nonrelativistic physics with this expression of relativistic physics with that pressure when you calculate in relativistic physics the amplitude to go from 1 to 2 was called the propagator call the propagator for the particle grow from 1 to and you do some fancy things you for example consider prophecies where particles coming joined Split go out a similar kind of action is defined in terms of the action 3 each 1 reviews which we call residual segments you the action together for each segment and then you don't summer just a single trajectory use some or all of these trajectories on top of everything else you eat integrate over the points where the trajectories came together so you don't say kind of train give your find diagram and they didn't make the whole thing even more complicated use some or all find diagrams for due process and that gives you the amplitude not for a particle to grow from 1 point to another but for a pair of particles to come in and go out but that's a basic set up of corn feel of finding by find path integrals before we go into string theory Chu trek that's physicist use over and over again it's a cheek to cheek can be justified in some special cases it can be justified far different turns out to be more easily justified in string theory than in quantum fervent field theory but I'll play with it but this kind of integral what this some over trajectories it feels out old trajectories summer lipstick nuts but even concentrate on the nonrelativistic case feel about trajectories which goes like this it feels out trajectories which got like missed your substrate trajectories feels out from wire while saying trajectories are some of them somehow more important than others of course they are the ones which correspond to the classical motion straight line ones must be the most important 1 otherwise classical physics would make sense but when you look at the Grand here he it I learned anything always has magnitude what warned so whatever trajectory as a whole have equal numerical weight they common with different phases the different phases can cancel each other if you're a ending up numbers on the complex plane and all those numbers happened perhaps unit magnitude that doesn't mean it can't canceled we can give it can cancel derives tore real number they arise imaginary number the walls of things but each 1 of their contributions is as big as every other contributions in this case on normal definition of an integral doesn't make it doesn't converge it doesn't converge because the trajectories which are we
wildly way out there correspond some very very distant trajectories don't have smaller and more ample amplitude as the trajectory gets more more wire so strictly speaking these happened to growth don't really exist as well studied mathematical objects OK so how we deal with them well 1 way of dealing with them is to be extremely careful defined integrals and extremely precise way after all sorts of reservations and things up to hard even for people who do do think about really hard stuff as attract the truck in the Czech always works the trick is 1 which justify it is as hard as trying to understand the original integral but once you justify it's a trick which always works always gives you the right value of intricate let's which will show we take let's take this case he wrote Mr. case you change the tolerable and how do you change in their river recently you will event on wearable called S & S is just a number times how will choose the number Alpha later convenience it's just a change of variables misses integration variable incidentally across from Taiwan power to where talent have 2 of you and points of the trajectory but which is variables this asks we shouldn't confuses s would be action I'm sorry you called that but scholarly actual give the action another name in order to avoid confusion you right little adds a little at Willis is by definition are fatigue and what is office would be 1 of the base will pick it later I'm so let's see what we get DTA our that I want no I think it would be smarter I'm going to set child equal plans s powers you go about band that I don't have to invert the equation 5 give this gives us it will land from S 1 and S 2 endpoints now are we end points of the essence of bullion points of the essence of course just Alpha transient point values we How integration OK now we have D. tell How Powell is out yes and then we have DX what's the X by D. Powell in terms of the expert ideas wary power use sticker oftentimes assets all 1st becomes beat X Y B S squared I won't Barbara regularly in the indices field but world laughter square feet for each How I have to put out estimate is we were about 4 square without square downstairs but see action Alford downstairs but nor Nauru we do with action we put up in into the exponent of the party could during this feat the 1 half I suppose authority right but just for the 1 half hour he would keep it out of our way the I of course is just a constant victim be brought inside and I all right now now it's complete nonsense for Amistad needs In major justification to do this track but will units if they were Jews debate light side but see IOI is is warranted I think I want be minus I that sounds right eye over on a big or Alfred equals which is to be minus His becomes on a over minor side which is also called like this 1 and the hall into Idaho exponential magically loses its because he integral might miss that the minus sign from my own mind aside 1 day after D X by D squared X Billy place that we have to build a temple half that remembered in points of the interval here on not really real values of us would not pretend that Escambia a real quantity of this In the book be treated as a real integral even tho we end points of integral I'm not real numbers why are they real numbers because how is equal Midas RAS and the end points of the original in Kabul were real values of time Talbert let's forget that but just call from S 1 test and later on will remember that as 1 2 or imaginary value later on momma just leave it this way now they integrate Over all trajectories or trajectories this is much better
than the original integral the original integral no matter how while the trajectory wars be amplitude for that particular trajectory woes in magnitude 1 now let's take some wild trajectory and stick it in here at the trajectory is Wyo goes off and extends after some Lauren Joe spatial separation incomes back in as good places would the X by B S is a very large trajectories while in any form whatever maybe just because maybe it doesn't go very far images wiggle extremely rapidly plus OK TX by DES is a very very large Castle large for maybe goes off then comes back along still BX by IDS will be very very large will long be orbits word goes out from the back of any kind of trajectory far from just simplest trajectory namely a straight line we action as a B huge WTO which comes in with a negative side each of miners are large numbers of small number sold in this form wild trajectories don't contribute much only trajectories near the Strait trajectory our sport this is a form of happened to grow that is used in statistical mechanics call the we're in the girl Is was not was invented before the final happened grow it's perfectly convergent mathematicians nor have defined the mysterious way and the quantity so this is what this is what we are always doing we always making his trick of turning a time variable side by making it I kind of kind of kind variable and then using Indian In quotes now what's what's a trip at the end of the day this whole thing is gonna be some functions of the endpoints of the interval no value which not paying idiot attention to the fact that the importance of the interval are imaginary numbers but then a hefty were evaluated using the right word is analytically continue with were extrapolated to imagine values this is something that we know how to do it so straightforward procedure if you actually evaluate these things for the experts there analytic functions of and you can extrapolate Demel analytically continue them the desired complex points this idiot only real way that physicist ever really do have the girls when you hear people talking about path integrals easily just talking or actually trying to perform and they're doing it in this way turning to ponder site but but I would tell you that once and for wrong because we do have string theory shorts counter string theory now say the same kind of set up but now a particle is not appointed to spring it's world history is not world line it's a surface that could be open strings in which case it to serve as a rebound and remained a red hurt change its lane and changes or awaits embedded in space but it's a ribbon closed strings of Sourav in that told court world tool is recall the world sheet that's motion of a particle let's talk about its actions were interested action kind question we're going to ask is giving the location of all the points of string of initial kind and some later time watch the amplitude to go from the initial state finals what's the amplitude of his set string up in a certain configuration and let it go you would
find it at a later time at some other configuration but now the location of a pouring on the world sheet is a function of X but 2 variables 1 of them is a kind of kind variable along the sheet will call it's not real time it's real time no more real try the house variable appeal more like a proper time analysis the parameter it's just a parameter you take sheet you call One NYTimes if we have to coordinate stigma and power you're right X of columns we've done that before it's a function of 2 variables if you know X new each segment how you know where that poured strenuous forgiven the whole history of the stringers X newest Sigmund out foresters again you construct an action in Ukiah struck action and this time the actions of the media generalization of that but with 2 independent variables s and go from Motorola murder trial label was starting to competition labor will start the finish what is right what is integral how these sick man it's an integral over the two-dimensional world sheet on DD X new palace quarters among Oh yes might this be X New by DEC much coed rabbit right next to my count squared minus nobody Sigma squared that's the action for the the notice this is the action of a field moving in Kowloon Sigma space it's just the action of the way goes moving up and down the house in our house said but it is also the action for each tool moving through space time a comes from the fact that power was comes from the same place OK comes from kinetic energy potential kinetic energy potential energy this is kinetic what time derivative misses potential energy so the fact that the action or grungy and as Microsoft might be worried section you exponentially 88 utility alright the you integrated all all possible surfaces madam Trevor draw a complicated tried and failed on new integrated or summit of all possible services incidentally is a classical version we don't do the path integral which just tells you if you start string in a certain fashion in a bid in a certain fashion however has to move in between the satisfy classical equations of motion go from initial viable way interested that which inquire mechanics this is the amplitude for the string to go from an initial configuration profile configuration axis set up that these like after all is said and done this pretty much be defined could to be the definition of string theory well 4 if a generalize the idea this this a puzzle I'm not interested in the string going to a string but 2 strings going to to strategy was to strings going to 3 straight the newly generalized are starting to strings ending with 3 strains and drawing possible services that can interpolate in between there's a surface that interpolate between earned there are other surfaces which interpolate between them with different apology to compose a hole in here perfectly
legitimate but let's concentrate on one's without holes in the moment you ask the the question What is the amplitude to start with 2 strings each 1 phone given configuration and end up with 3 strings also particular configuration the answer is the same the answer is he I in the world over all surfaces which connects the initial state the final state Vitaly I. the growth of all surfaces action for Webster this was lecture on 1 the Yanks wrote this is a point by point electronic you would dealing with it by finding was before eBay heard of it theory you would that you would never sent you just have Our we acting Maybeury asking whether In some limit or other this reduces the yacht now a the question the answer is yes our as long as the electrons very much lighter than the string scale and as long as the EU energies of the process is a low then yacht these things will love will reproduced by the diagrams and not connected that would take us the borders were OK so that's but practically definition of string theory incidentally if you really needs some or all surfaces that connect including the surfaces that have holes in the various kinds that's pretty much all the analog of some all finding diagrams with all possible topologies all possible number of loops so they have in 1 simple expression 1 simple idea some over surfaces you have something which is as complex or a layer of complexity perhaps word much simpler by bag in many ways lot but it contains as much the same kind of information is the sum of all funds bad OK but let's forget these calls from Mommy concentrate on the structure of this path of patentable for string again it's a monstrosity for the same reason wildly varying surfaces contribute as heavily as Street services 4 of the interval doesn't converge in any meaningful sense because again of surfaces which stretch out their office and Korean comeback are no less numerically large Indian amplitude then a nice simple tools facts about means we've got to do something to make this into Goma accepts you can do rule what a series mathematician would do would be to try to define a new kind of integral people of study that will you could do a physicist would do now however poetically continue however could continue last refuge of scoundrels we are going to be scoundrels what deal not we'll Malloy about sigma segment who is not problem actually power which is our problem as we will see it let's still the same thing let's let Powell top I count and that the call it acts amount mechanical SO just get too many variables of district we know make a transformation of variables From Powell thought I can say trick when you do with your pick up and ii from here and you will pick up 2 wines from here you won't pick up any eyes over here so it's each time you see How you were you right White House really count was becoming c est but don't want to introduce a over a number of variables so I'm just gonna say how co-star might change variables like then the 1st thing that happens we pick up and Iowa he added that's going because Riverside but they would pick up to wives down he but what is miners so that isn't a minus here say is loneliness he didn't get rid of the he no unfortunately I been really mean it when I see how ghost I'd tell how goes to my side in which case we have my side here saying this for the particles case now that we have we have something well I don't know why are actually to many miners I'm lost control of the calculation last now the case is 1 best form but had my size works also on the plus side headed for vote now Joe what you want it With want yet so let's get this work but that our how was my boss might decide tower the reason I want the plus sign minus signs here is is that allows me them will remove the minus signs here and put out here know why is it that a good thing do if it's a good thing do because what appears in here is positive the while the services the Bure it's gonna be even if the surface doesn't flocked to wait out the far distances but it does vibrates a lot then the XT cowardly or Whittle's too much in the way of the XT signal large vibrates too much BXB power will be large stretches out 2 office and century both of of them are gonna be large and positive so 1 good thing of this Estrada's babies both of the same starting and both past and then Ito mind a positive number was small or large positive number so was as it does saying they had given the particle case it suppresses the wildly varying surfaces and allows you will actually calculate I tell you there's another form of string theory where would you put in for the action is more complicated but quite beautiful the it's actually from both the particle and this during are other forms of the reaction which are mathematically harder to deal with but sense geometrically very very high remote motivated for the case both the particle what you put in the proper path point as the action the action is just a proper Pampling between initial point final point the proper path itself and year of the string would you put in is the space time area of the world she that's another set up another for Moor we could do it gives you the same answers and there but is more complicated if you ever come across the turned them will collection mambo Goto action the action which just stick in the action be area Of the leader of the world's to their former sometimes called Holly a class action or have nothing to do with Polyakov who was invented by me which Polyakov
constantly keeps telling people but they will likely decrepit LIPA credit for a gay rights quota political factions action did something else which is related to this but armor but that but that nothing he did which is closely related to this as some other factors in it which is also quite beautiful which amid the Polyakov Koppers Co believe election it was a mathematician who died long long before you in classical mechanics so don't ask Sam reason that Koltchinskii brains called do brings Brent oil at Rasheed Wallace they were not due later boundary conditions on ordinary strings warm turned not that's correct OK that is the idea of the captain and the role for string theory now want to explain some mathematics so trivial matter trivial the West mathematics many you before but but young would look at this action and this quarter to wanted some white or other things that's invariant while the beans of invariant what kind of transformations in particular of Cigna and how into new variables is Wyoming interest in the well I never told you exactly what the choice of Sigmund Cowwarr I didn't tell you have to sit around town 1 here has drawn anyway we please surely the Berryessa can't be the same any choice of certain power must be some rules about what can coordinate 2nd used on the world sheet but are those rules rigid because to use these particular coordinates all resistant variants which allows you to arm the however a variety of different set of coordinates which will all give you the same answer as the classic problem of the invariant around but oddly and variances of this Lagrangian but particular what are the invariants is with respect to coordinate transformations of the sea surface of the sea how parameters so that houses labels of point so they don't have any independent meaning other vendors labels appoints this 1 is called Georgia's went call Fred is would call Harvey clearly cancer can depend on whether I renamed the points some Nova and aware of your case so what in variances that that allow so we look at this action what kind of what kind of transformations we make let's look at Nauru more detail here it is being acts by fact instead of looking at be acts in itself let's look at the equation of motion which would come by varying Sachin why it's just it's it it's as good as looking at the action the equation emotions that would come from varying this action would look like a wave equation this is basically reaction for a wave equation with 1 exception reports exception was signed here you have to pass what a wave equation a wave equation would be a 2nd how squared minus 2nd ex be stigma squared equals 0 now With think the equation emotion coming from there from Arabs from from this will X is some field on the palace signal coordinate space you know the name of this equation all the way the course Philip Glass's mature plus equation the two-dimensional what these 2nd the something with respect to 1 variable squared post usually is even differ form you see the variable X will be replaced by some function just as functions and Palin's Sigma would be called X and Y to equation like these 2nd half would be 2nd wife actually be X squared was 2nd by Y squared equals 0 in 2 dimensions in 3 dimensions you would get another 1 B 2nd foray by busy square I'm talking now not about physics give up a formal structure equations he recognizes equation yes by a of life OK again guys your party right of vibrating drum also has a time variable so I want you vibrating drums but you're not completely wrong lumber is equation the electrostatic potential except the electrostatic potential you have a charge density on the right side as Colo a total a plus equation and now it's all last equation dimensions except now the variable fire is replaced by X and the Triple X and Y a replaced by counts but otherwise mathematical structure equations the question now caution yeah I 1st year 1st derivatives yacht for your 1st day of his action it gives you 2nd rivers in the equation emotion just like having X thought squared connection to X bubble not quite so but request whether I work with the action or the equation emotion as a matter of you get a 6 1 half of I just like form better because tells me illustrate point right now let's see what is the 2nd derivative 2nd derivative of acts think of but how axis was being discreet there are just but to give you a geometric picture of what this equation me think Powell axes it as being discreet with little separation epsilon between neighboring point what Is the 1st Direct a 1st derivative it is what causes . 1 with 1 2 and 3 Of 1st derivative is closely related to the difference of a quantity at neighboring point so the 1st derivative it would be like X 3 miners extra tool you divided by Epsilon this is not important survive by epsilon epsilon the distance between the points new let the Epsilon xeroma and that gives you the 1st derivative let's not worry about the Epsilon right now that's not the point of 1st derivative it's like the difference of something and neighboring points if that's what 1st derivative is what is 2nd derivative a 2nd derivative is difference of 1st derivatives I saw 2nd derivative x 3 minus sex to rule as 1st derivative evaluated between these 2 points my the 1st derivative evaluated 1 step over minus X tool minus acts of 1 that's a analog disagreed analog of the secondary to its 5 x 3 x 0 3 plus Exxon minors twice X to SOLAS it it's the some of the quantities on the outer edges minus twice the quantity and that's a 2nd let's look at the thing which occurs in the last equation here instead of taking the taxes to be discreet let's take power signal planed to be discreet the Council the plane is discreet now
unless it would geometrically identify with this equation but Goto . 0 India point over he the 2nd derivative of X with respect the how is exit this point plus exit this point minus twice the sent to adjust to get out of labor . 1 2 3 4 and the center will coal-fired so these 2nd ex by D. Powell square that acts of 1 plus 603 twice the fire right about be 2nd necks with respect similar squared board Iwo sorry I make a mistake you OK pennzoil which we think of how I guess I have this picture here How willing horizontally and signal vertically don't usually think of it that worried but the so exits at the 2nd derivative respected is this 1 plus this woman is this what about the 2nd derivative respect to stigma that's gonna be this 1 pass this 1 minus 26 of 5 so we did edit plus exit to 4 sacks of form minus twice sex of 5 0 that's a bad makes a 1 plus 6 3 plus acts of war for sex of Joe 6 of 1 3 joint so we get the sum of the exodus from point surrounding acts of 5 minus 4 times this some started 4 times X at 5 equals 0 Netherlands it was that Texas 1 4 sex of 2 plus X Our 3 plus extra 4 equals war next at 5 X at 5 4 you're not later say is it excess 5 is the average of react to fund that's what low-cost equation says it's true in higher dimensions were also became a focal a blast equations says that Eds point if surrounded by a square and even know this thing is on side it is a square a square a small square infinitesimal square but that if you evaluate be field at the some of the some of the field at media at the corners but the field at the center average the quarter now the
square doesn't have to be oriented this way it's well known that this equation is invariant under rotations Of the Sigma so you could say it even more geometrically take any square and a square oriented in any way any small square was of course any a trooper consider possible square for a small infinite Cosmo square field at the center it is the average of the field quarters that's invariant way apart from the fact that you have to think about limits makes a little more complicated apart from that limiting procedure that you have to do that's an invariant way to say a word report plus equation is very awake describe the equation of motion classical equation emotional where OK so now I can tell you exactly what the invariants structures or plaster waste or of this action it's the class of transformation of the signal Powell plane which take every infinitesimal square into winning possible square if under coordinate transformations of the segment play squeezing mappings of the Sigma power plane it takes every small square until the small square then will pass equation will be will be unchanged in Newport so every some no coordinates courts and the prime which is a function of Sigmund pound and how prime which a functions of how it is true that small stigma power square maps in tour the stigma planes repel playing map of Merrill who the prime power most mappings will not take squares square the thick squares or something else but as a special class of mappings which take squares the square you may know the names such mapping on former Womack Khan formal map now express it another way if I have a transformation which preserves angles at all angles preserved there would be a conformal transformation it will take squares to square White well 1st of all let's take me on the diagonal was a map of the the prime help by plane nobody little short segments oriented and some of action let's suppose that all the angles reserve on the map and the angle between 2 curves becomes angle among the signal prime prime place then 1st of all orthogonal angles he will go toward firing line he is not true of any old transformation it's true on some classic transformation the angle preserving transformation Bill because something else supposing that so that it had the property of picking her perpendicular isn't he perpendicular but stretched things out in in 1 direction member warned me that kind of elongated hiding out of a square than they would be other angles which warrant preserve these angles over here would be changed this angle at this corner would not have this is a 45 degree angle stretch the square and some way it would not remain 45 45 degree angle conclusion angled preserving transformations preserve little square take little squares squares now it wasn't that big a patient locally it's only a rotation but a verdict not they can bend the quarter snorts not to accept as protection but remember were only demanding that infinite has small Our squares remains the a square but not say without In using the word infinite has more if the transformation has the property that be angle between you and me intersecting link pair of curbs on the signal power play maps intimacy angle on the part help for then there things come can't formal mappings preserve not preserve angles preserve all angles for sufficiently small shapes was officially small shapes Yukon think of the curves being straight lines and seed approve a sufficiently small shapes but themselves all rules Hoosier
strategies use or not sizes among reserve a small square can to a bigger square Romania a mail drop in our name and sigh I sizes can also be a veil we're talking about the rotations and translation right that's are much much more classic transformation talking just about angle preserving transformations and they can expand and contract chip in fact can expand and contract differently in different regions of the plane where he received my get stretched to a bigger states already here the shade might get squashed to was more now is that could save the corpses not from some formerly flat but we backwardness system being flat Asian hurt I don't know where is this trick of conformal mappings you well any projection of a map which is intended to to preserve the shape of Ireland's compliments salt what's more ones but doesn't care about the sizes of things that's a conformal mapping so the relationship between different maps different projections both of which preserve shapes but not sizes is always a conformal yup last week he said yeah I know this is you had to that that this is the biggest blow in that it now is says it yet Rural Charter dressed me right narrow worker relationship is 2 ships I was gonna tell you a little bit about analytic functions but combat coordinate transformations which are defined by analytic functions of complex variables are angle preserving we will come back you're but but Oh incidentally there is 1 Our situation where angles I'm not necessarily preservative curious 1 arm you can't take any figure with a boundary but say square a big scoring not until has more square a finite size square but 3rd up I can't formal mapping took NEC feel like but a particular you could match that 208 this system of square Ucon Africa delay like that the winner you said that they have to preserve angles this doesn't look like a provisional 1st them all are it only preserves infinitely infinitely small shapes you can map this formal mapping but 2 the corners will not go to the corner the corners Shia will perhaps go to these points but for Kirk Seoul could come from Rabin's 30 you commit any shit for the boundary any of the shape of the same amount this is fear about our mappings and inside the region any small shape will map to shape of the same kind of any small shape will map which small sin kind In the interior of so this is something know that you can do some things were changing this to the shapes of boundaries of OK having so that now will become collector what point last time we talk about strings bleeding joint I drop picture for you where 2 strings came in and I indicated that by making us slit the Caribbean maker slipped Elmer until last like comes to an end point those at 2 separate strings joining at this
point and making 1 long string and they are continuing again after exports again like that there's a 1 parameter family of such splitting joining was characterized by the time interval was how interval between his point in that for me and if the calculating the scattering amplitude you integrate over you get some integral and that hinder will give you the beta functions or the beta function in its young and with food her your we wrote to describe euphoria last time love and it gives you the scattering amplitude an important factor but not really terribly your OK what's going on here what's going on in here because we have a two-dimensional space with quotes Cigna Powell should only have each point in the space we haven't X not as 1 expert several acts of family of axes knew of stigma and and they satisfy some kind equations and that what kind of equations less satisfied they satisfy the law plastered equation not actually technically they satisfied we equation but if you do this trip and carry out the path integrals by living how good might then be X is in he is satisfied with plaster what does that tell you it tells you that you can map have shape for you we have 8 a piece of playing it goes out to infinity along he ever think of as being slid down here finger was being down the center here what's the apology of this region here a player of the policy of this region the topology of this region is a struggle over some special wiser circle on let's start the center circle and work out a series of circles the bull's-eye what are the center of this work hours I can do this in such a way as to cover the whole space but a sequence of close curves which are isomorphic the circles of the words I've taken the bull's-eye and performed it but without changing its topology I've changed into this double strip car with the these points over here now got got right the some exceptional warning some exceptional warned exceptional porn on the infinite and points they are a but you can go this mapping is mapping exist and you'll have to remember that there were some exceptional warrants which have to be where ethanol what do with those exceptional points for the answer is that we the that Beyond what Corless shape but to a accord slipped the slit stripped the doubly slit strip but I'll OK a mouthful the what no no no no cut down the middle so it's quite a bit work they will all we can week and what the sport's mandated that right and good except that it's the infinite support OK that's not much better than that of the minutiae of the boys would strip him sports Band-Aid the infant Boyd's Band-Aid can be can't formerly map come formally map the the circle but earlier this good there's this circle is the interior of a circle sorrel that leaders in principle we can take the wave equation we can take the class equation or the Chen's of the action of the year of the strained of patentable for the strain the entire set up and rewrite it In a former it would be Exs is all live on unit circle Hey where the X is all lived on the unit circle the something special about these 4 points here with particles he and went about her eventually tell you what a special thing you Camacho Bahia cluster African Paducah map he'll without changing the form of the equation it's still just pull up last equation of class equation for the extras in here it is the transform form of equation them union here OK that's because very plus yacht because the conformal invariant will sequence some mucus set up the stream the path integral in Norway by taking Euler guests thinking of a set of Exeter's which women unit and integrating all all ways all Exs that live on this universe or not think you understand Texas segment segment how are not on the other board Band-Aid but they're the writing the action has the same kind of action and path integrating over the X is in here with some special things going on at least 4 exceptional our I think I won't get into work there accept this nothing very complicated about How will we will get to it will certainly get away but technically or or in the jargon is you have to put some on vertex operators he pored over here but will come but you could also strokes thousand the other other shapes you could stretch that I shape which required myths went off to infinity star looks actually more like a scattering process and they things you could stretch that up to now but their break obstacle Bert right now and that thing this
problem for this problem this problem I mean the problem of calculating scattering amplitudes you there is something but you have to do our at these infinite points here set up and specify the initial and final states of the particles comedian mentor if particles tight money the ground states of the 1st excited states someone errors in here the sum input put on information
but above and beyond just saying that because class equation in here that input of information has to do with something or other that we do lot out I'll be the NUC amateur draw out just some input and media and flaps of the Band seemed input has to be put into this diagram have to say what kind of particles were as their momentum not tell you exactly how do that at the moment I will tell you eventually but some operation that you do on each 1 of these points 2 In Jack In a certain amount of momentum and some information about particles to keep them right now where exactly do we put such as those 4 points in this diagram it's clear we put them we put them out there of and flaps of the year the mandate and this diagram is less clear and I will tell you what it was correct and that thing just to hear you forget it forget for a moment here you have some ambiguity for example you could rotate could do some of the things to it but you also have enough ambiguity that you cannot arbitrarily choose that does really only 1 parameter in the relationship between the 4 points on here 1 parameter all the other looks like they're really several parameters euchre arbitrarily picked by rotating angles a position of this point but then they would be going door distance to hear the angular distance to hear the angular distance the year saying it would not are you could find conformal transformations which will move these the boundary points around the various ways can result there is really only up to a conformal transformation is only a 1 parameter family of such configurations could describe the parameters by saying all arbitrarily choose left-right symmetry that's a choice now also chose up Lublin vertical cemetery packages this pulling over here and said whose decided to make a diagram of left-right symmetric Gisela point here here and although he had completely symmetric way but that still allows a freedom here's 1 extreme situation it is another extreme situation want determines where you put the points where looks like this like this all like that the answer is the time interval between here and here which is also a 1 parameter family of figures the time interval really strictly speaking the ratio of the time in the world the the separation wall here all the time interval between how would between here and he is the 1 parameter the government's where you put these points you have they integrate over just delete integrated all time separation part of the path integral with integrated over kind separation between joining in the splitting that translates until 1 parameter integral all over it we can chose in many ways but in this case With this particular set up its own 1 parameter in the world all over how we choose these points my fingers on big enough to stretch stretch the configuration here we had 2 of us up here I would form a figure like back you would move your for that amount go your way that the girl was still to be done such that integral when you work out the path and the gold and then a over the positions of the points in the right way that's what gives you the beta function in this language the beta function comes from integrating the positions of the injection points corn injection points positions of the injection point over 1 parameter family they're O'Toole extreme situation 1 extreme situation is this the other extreme situations like this extreme situation the 2 input points in the last up close together In the other configuration over here laughter this 1 and this 1 of course together now this this these 2 extremes we must translate to some extremes on the mandate does the extreme of where the 2 incident particles appeared adjacent which other the outgoing particles appear adjacent to each other that is kind formal mapping of the configuration with kind interval here is very very long and so it looks like these 2 points a much much closer there distance between here and here that we show like the mapping from Band Aid the circle on the configuration where intermediate state last a long time it's situation that manifestly looks like the particles coming in forming the Stringer vibrates many vibrations and then goes off again that's slim what seal and well yellow that is exactly and were time interval was very very short the intermediate state last over a short time Ford bad and then the splitting occurs and that's Best limit In these integrals in the various and the was are you do what you could see from this picture but there's some
symmetry that there's some symmetry From this Petrovic year it's hard to see that there's any symmetry between these 2 extremes limits to physically public different pictures won an intermediate state west along time and the other in intermediate state last loss short track was awkward as much symmetry but from this picture he's seen that the that somehow there is a cemetery but they must be related to each other in some ways how they relate to each other however that's not the point the related by the interchange of what we call last time S variable and the key variables remember would be as variable was it was some of the foreign moment they're coming in square on the center of mass energy square free labor leaders with Mom and Dad the S variable Kate warning Case tool K 3 K former where then the Mandelstam variable Mandelstam as variable was K 1 plus K to squared and corresponded to to the square of the center of mass energy Fred Segal Mandelstam as all this is not the action is not any other SO used tonight and a mammals stem as bearable and it's also eagled the square of the center mass-energy December on the other hand it's quite clear that we've done what we've done in going from here he we will it be changed K 1 plus cake to square of C COBOL energy coming in here with was a key 1 plus K 3 Katie warned Post race and there so we called the key variable that's such center of mass energy that some combination of center of mass energy bodies center mass square miners and squared and I think 1 minus the cosine of the angle scattering or something but this just K 1 plus K 3 square moment their account is coming in in this form aides manifestly clear but that these diagrams symmetry between U. S Channel and teacher its also cleared but if they have if they describe particles forming a indicate in India's Charo particles comedian forming became they must also describe processes Warner the 2 incoming particles of replaced now by 1 incoming 1 outgoing particles exchanged not not the creation and annihilation of a particle but the mission and reabsorption of the other side of the diagram so what that's what was so fascinating about these structures is that 1 structure 1 simple structure 1 in the world but 1 basic principle Was accounting for both the S Channel and the teacher particles as cannot particles were somehow associated with lawn diagrams like this teacher worms were short war but does we not have to interpret we just say in Kabul symmetric under s goes the sir whatever was there in the form of as particles formed by cable and Quito coming together there was also feature objects which were formed by warranty 3 coming together that was cold do our that was the 1st duality duality between U.S. channel and channel now these days it called channeled 3 jails in this case means you are clearly it's called channeled or our was the 1st of at 1st it was a miracle at 1st thought about the terms of Band-Aid diagrams it was a miracle that that a was a symmetry between national Peter was very quick there was however was realized that this was because of formal symmetry plus and make you could map it was more of a symmetric situation but 1st at 1st sight it just look like America the trouble with understanding miracles as always has the same effect not only do you may now understand it but no longer got new went into brought us a lot of them I thought they well I yes you could draw back the other way you could on time for assault guns could well yes well not time comes in here the Ukraine that case should could not afford plan around there horizontal you have to say this was deep promote passed over here and this was deep promote future over here but you could on their salaries that a sign that the deal later got a nominal some kind together you have to integrate over degree over would you trying to say is Red were there probability fruit became is related to a couple of need a case nor what you spoke that just means you have to put into the integral the coupling constant knowledge a these diagrams coupling constant just comes in as a multiplicative factor but OK I think that's enough for tonight for more
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